Green's Theorem

Let C be a positively oriented, piecewise-smooth, simple closed curve in the plane and let D be the region bounded by C. If P and Q have continuous partial derivatives on an open region that contains D, then

Discussion

Take for example the figure shown below. C here corresponds to an oriented close path or a path where the starting point is the same as the endpoint. D here is the interior region of closed path C. Greenís Theorem transforms a line integral at a closed curve C and a double integral over the plane region D bounded by C.

In stating Greenís Theorem, we use the convention that the positive orientation of a simple closed C refers to a single counterclockwise traversal of C. This means that if C is given by the vector function of r(t), a \leq t \leq b, then the region D is always on the left as the point r(t) go across C as shown in the figure below.

Example#1

Evaluate

\int_{C} x^4 dx+xydy

where C is the triangular curve consisting of the line segments from (0,0) to (1,0) , from (1,0) to (0,1), and from (0,1) to (0,0) .

Solution. We can actually solve this by having three different integrals of the functions from (0,0) to (1,0) , from (1,0) to (0,1), and from (0,1) to (0,0). Here, we would like to show a different approach using the Greenís Theorem.
We let P(x,y)= x^4 and Q(x,y)= xy. By Greenís theorem, we have